£ 128 ] 



XV. On the Method of developing an IncommensuraUe Trac- 

 tion given in Colson's edition of Sir Isaac Newton's Fluxions. 

 By J. R. Young, late Professor of Mathematics, Belfast"^. 



THE paper on Incommensurable Fractions, inserted in 

 the last Number of this Journal, suggests a modification 

 of Colson's rule for expeditiously converting such fractions 

 into circulating decimals, which will, I think, be regarded as 

 a slight improvement. 



The fraction chosen by Colson, to illustrate his process, is 



— ; which he developes, in the usual way, till he arrives at a 



remainder consisting of but one figure : this remainder, with 

 the 29 underneath, is then appended to the partial develop- 

 ment, and the true value of— , consisting of a certain number 



of decimals, with a supplementary fraction, is thus exhibited. 

 The whole is then multiplied by the numerator of this fraction, 

 and as many additional decimals are obtained, together with 

 a new supplemental fraction. 



The whole row of decimals now furnished, with the new 

 fraction appended, is, as before, multiplied by the numerator 

 of this fraction; and the extent of the row again becomes 

 doubled, and another supplemental fraction presents itself, 

 the numerator of which forms a new multiplier; and so on, 

 till the circulating period is completed. 



Now, in certain cases, there is much inconvenience in thus 

 changing the multipliers at every step; for although the mul- 

 tiplier with which we commence may be but a single figure, 

 the subsequent multipliers may, some of them, consist of two 

 figures, or indeed of any number of figures which do not cause 

 the multiplier to be so great as the denominator of the original 

 fraction. 



It is an obvious inference from the paper referred to above, 

 that whenever a convenient multiplier is reached, we may, if 

 we please, keep to that multiplier to the end of the process ; 

 and by so doing, we shall add on the same invariable number 

 of decimals at each step taken from the stage at which our 

 multiplier vvas selected, namely the number of decimals fur- 

 nished at that stage. If, as we proceed, a multiplier still 

 more convenient offers itself, that at first chosen may be re- 

 placed by it; and, as before, new rows of decimals will be 

 added at each subsequent step, the number of figures in every 

 addend being the same as the number in the row which sup- 



* Communicated by the Author. 



